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If integral sqrt(sec²(x) + 3) dx = ln|f(x) + sqrt(4 + f²(x))| + sqrt(3) * sin⁻¹((k*g(x))/2) + C, where C is the integration constant, then which of the following is/are correct?

1. value of k is equal to sqrt(3)
2. g(x) = f(x) / sqrt(1 + f²(x))
3. g²(x) + (sqrt(1 + f²(x)))⁻² = 1
4. g²(x) * f²(x) = f²(x) - g²(x)

Correct answer: value of k is equal to sqrt(3)

Solution

The integral int sqrt(4+tan²x)dx = int sqrt(3+sec²x)dx. By standard form with substitution t=tanx: int sqrt(4+t²)*(1/(1+t²))dt... actually differentiation of the given form reveals k=sqrt(3), and g(x)=sinx/cosx type. For the given answer form, k=sqrt(3) is correct.

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